In statistics, a percentile (or centile) is the value of a variable below which a certain percent of observations fall. For example, the 20th percentile is the value (or score) below which 20 percent of the observations may be found. The term percentile and the related term percentile rank are often used in the reporting of scores from norm-referenced tests.

The 25th percentile is also known as the first quartile (Q₁), the 50th percentile as the median or second quartile (Q₂), and the 75th percentile as the third quartile (Q₃).

Definition[link]

There is no standard definition of percentile,^{[1] [2][3]} however all definitions yield similar results when the number of observations is very large.^[4]

Nearest rank[link]

One definition of percentile, often given in texts, is that the P-th percentile (Failed to parse (Missing texvc executable; please see math/README to configure.): 0 \le P < 100 ) of N ordered values (arranged from least to greatest) is obtained by first calculating the (ordinal) rank

Failed to parse (Missing texvc executable; please see math/README to configure.): n = \frac{P}{100} \times N + \frac{1}{2}

rounding the result to the nearest integer, and then taking the value that corresponds to that rank. (Note that the rounded value of n is just the least integer which exceeds Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{P}{100} \times N .)

For example, by this definition, given the numbers

15, 20, 35, 40, 50

the rank of the 30th percentile would be

Failed to parse (Missing texvc executable; please see math/README to configure.): n = \frac{30}{100} \times 5 + \frac{1}{2} = 2

. Thus the 30th percentile is the second number in the sorted list, 20.

The 40th percentile would have rank

Failed to parse (Missing texvc executable; please see math/README to configure.): n = \frac{40}{100} \times 5 + \frac{1}{2} = 2.5

, so the 40th percentile would be the third number (since 2.5 rounds up to 3), or 35.

The 100th percentile is defined to be the largest value. (In this case we do not use the above definition with P=100, because the rank n would be greater than the number N of values in the original list.)

Linear interpolation between closest ranks[link]

An alternative to rounding used in many applications is to use linear interpolation between the two nearest ranks.

In particular, given the N sorted values Failed to parse (Missing texvc executable; please see math/README to configure.): v_1 \le v_2 \le v_3 \le \dots \le v_N , we define the percent rank corresponding to the n^th value as:

Failed to parse (Missing texvc executable; please see math/README to configure.): p_n=\frac{100}{N}\left(n-\frac{1}{2}\right)

. In this way, for example, if Failed to parse (Missing texvc executable; please see math/README to configure.): N=5

the percent rank corresponding to the third value is

Failed to parse (Missing texvc executable; please see math/README to configure.): p_3=\frac{100}{5}\left(3-\frac{1}{2}\right)=50

.

The value v of the P-th percentile may now be calculated as follows:^[5]

If Failed to parse (Missing texvc executable; please see math/README to configure.): P<p_1

or Failed to parse (Missing texvc executable; please see math/README to configure.): P>p_N

, then we take Failed to parse (Missing texvc executable; please see math/README to configure.): v=v_1

or Failed to parse (Missing texvc executable; please see math/README to configure.): v=v_N

, respectively.

If there is some integer k for which Failed to parse (Missing texvc executable; please see math/README to configure.): P=p_k , then we take Failed to parse (Missing texvc executable; please see math/README to configure.): v = v_k .

Otherwise, we find the integer k for which Failed to parse (Missing texvc executable; please see math/README to configure.): p_k < P < p_{k+1} , and take

Failed to parse (Missing texvc executable; please see math/README to configure.): v=v_k + \frac{P-p_k}{p_{k+1}-p_k}(v_{k+1}-v_k)=v_k+N\times\frac{P-p_k}{100}(v_{k+1}-v_k)

.

Using the list of numbers above, the 40th percentile would be found by linearly interpolating between the 30th percentile, 20, and the 50th, 35. Specifically:

Failed to parse (Missing texvc executable; please see math/README to configure.): v=20+5\times\frac{40-30}{100}(35-20)=27.5

This is halfway between 20 and 35, which one would expect since the rank was calculated above as 2.5.

It is readily confirmed that the 50th percentile of any list of values according to this definition of the P-th percentile is just the sample median.

Moreover, when N is even the 25th percentile according to this definition of the P-th percentile is the median of the first Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{N}{2}

values (i.e., the median of the lower half of the data).

Weighted percentile[link]

In addition to the percentile function, there is also a weighted percentile, where the percentage in the total weight is counted instead of the total number. There is no standard function for a weighted percentile. One method extends the above approach is a natural way.

Suppose we have positive weights Failed to parse (Missing texvc executable; please see math/README to configure.): w_1, w_2, w_3, \dots, w_N

associated, respectively, with our N sorted sample values.  Let

Failed to parse (Missing texvc executable; please see math/README to configure.): S_n=\sum_{k=1}^{n}w_k

, the Failed to parse (Missing texvc executable; please see math/README to configure.): n -th partial sum of the weights. Then the formulas above are generalized by taking

Failed to parse (Missing texvc executable; please see math/README to configure.): p_n=\frac{100}{S_N}\left(S_n-\frac{w_n}{2}\right)

and

Failed to parse (Missing texvc executable; please see math/README to configure.): v=v_k+\frac{p-p_k}{p_{k+1}-p_k}(v_{k+1}-v_k)

.

Alternative methods[link]

Some software packages, including Microsoft Excel^[3] (up to the version 2007) use the following method, noted as an alternative by NIST^[6] to estimate the value, Failed to parse (Missing texvc executable; please see math/README to configure.): v_P , of the P-th percentile of an ascending ordered dataset containing N elements with values Failed to parse (Missing texvc executable; please see math/README to configure.): v_1 \le v_2 \le \dots \le v_N .

The rank is calculated:

Failed to parse (Missing texvc executable; please see math/README to configure.): n = \frac{P}{100}(N-1)+1

and then split into its integer component k and decimal component d, such that Failed to parse (Missing texvc executable; please see math/README to configure.): n = k + d . Then Failed to parse (Missing texvc executable; please see math/README to configure.): v_P

is calculated as:

Failed to parse (Missing texvc executable; please see math/README to configure.): v_P = \begin{cases} v_1, & \mbox{for }n=1 \\ v_N, & \mbox{for }n=N \\ v_k+d(v_{k+1}-v_k), & \mbox{for }1 < n < N \end{cases}

The primary method recommended by NIST^[6] is similar to that given above, but with the rank calculated as

Failed to parse (Missing texvc executable; please see math/README to configure.): n = \frac{P}{100}(N+1)

These two approaches give the rank of the 40th percentile in the above example as, respectively:

Failed to parse (Missing texvc executable; please see math/README to configure.): n = \frac{40}{100}(5-1)+1=2.6

and

Failed to parse (Missing texvc executable; please see math/README to configure.): n = \frac{40}{100}(5+1)=2.4

. The values are then interpolated as usual based on these ranks, yielding 29 and 26, respectively, for the 40th percentile.

Applications[link]

When ISPs bill "burstable" internet bandwidth, the 95th or 98th percentile usually cuts off the top 5% or 2% of bandwidth peaks in each month, and then bills at the nearest rate. In this way infrequent peaks are ignored, and the customer is charged in a fairer way. The reason this statistic is so useful in measuring data throughput is that it gives a very accurate picture of the cost of the bandwidth. The 95th percentile says that 95% of the time, the usage is below this amount. Just the same, the remaining 5% of the time, the usage is above that amount.

Physicians will often use infant and children's weight and height percentile to assess their growth in comparison to national averages.

The normal curve and percentiles[link]

The dark blue zone represents observations within one standard deviation (σ) to either side of the mean (μ), which accounts for about 68.2% of the population. Two standard deviations from the mean (dark and medium blue) account for about 95.4%, and three standard deviations (dark, medium, and light blue) for about 99.7%.

The methods given above are approximations for use in small-sample statistics. In general terms, for very large populations percentiles may often be represented by reference to a normal curve plot. The normal curve is plotted along an axis scaled to standard deviation, or sigma, units. Mathematically, the normal curve extends to negative infinity on the left and positive infinity on the right. Note, however, that a very small portion of individuals in a population will fall outside the −3 to +3 range.

In humans, for example, a small portion of all people can be expected to fall above the +3 sigma height level.

Percentiles represent the area under the normal curve, increasing from left to right. Each standard deviation represents a fixed percentile. Thus, rounding to two decimal places, −3 is the 0.13th percentile, −2 the 2.28th percentile, −1 the 15.87th percentile, 0 the 50th percentile (both the mean and median of the distribution), +1 the 84.13th percentile, +2 the 97.72nd percentile, and +3 the 99.87th percentile. Note that the 0th percentile falls at negative infinity and the 100th percentile at positive infinity.

See also[link]

Statistics portal

• Quantile
• Decile
• Summary statistics
• Percentile rank

References[link]

1. ^ Hyndman RH, Fan Y (1996). "Sample quantiles in statistical packages". The American Statistician 50 (4): 361–365. DOI:10.2307/2684934. JSTOR 2684934. 
2. ^ Lane, David. "Percentiles". http://cnx.org/content/m10805/latest. Retrieved 2007-09-15. 
3. ^ ^{a b} Pottel, Hans. "Statistical flaws in Excel". http://www.coventry.ac.uk/ec/~nhunt/pottel.pdf. Retrieved 2006-03-22. 
4. ^ Schoonjans F, De Bacquer D, Schmid P (2011). "Estimation of population percentiles". Epidemiology 22 (5): 750–751. DOI:10.1097/EDE.0b013e318225c1de. 
5. ^ "Matlab Statistics Toolbox - Percentiles". http://www.mathworks.com/access/helpdesk/help/toolbox/stats/prctile.html. Retrieved 2006-09-15. 
6. ^ ^{a b} "Engineering Statistics Handbook: Percentile". NIST. http://www.itl.nist.gov/div898/handbook/prc/section2/prc252.htm. Retrieved 2009-02-18. 

External links[link]

• Free Online Software (Calculator) computes Percentiles for any dataset according to 8 different percentile definitions.